reserve A, B, X, Y for set;

theorem
  for R being antisymmetric RelStr holds (the InternalRel of R) /\ the
  InternalRel of (R~) c= id the carrier of R
proof
  let R be antisymmetric RelStr;
  let a be object;
  assume
A1: a in (the InternalRel of R) /\ the InternalRel of R~;
  then consider y, z being object such that
A2: a = [y,z] by RELAT_1:def 1;
A3: y in the carrier of R by A1,A2,ZFMISC_1:87;
  reconsider y, z as Element of R by A1,A2,ZFMISC_1:87;
  a in the InternalRel of R~ by A1,XBOOLE_0:def 4;
  then [z,y] in the InternalRel of R by A2,RELAT_1:def 7;
  then
A4: z <= y by ORDERS_2:def 5;
  a in the InternalRel of R by A1,XBOOLE_0:def 4;
  then y <= z by A2,ORDERS_2:def 5;
  then y = z by A4,ORDERS_2:2;
  hence thesis by A2,A3,RELAT_1:def 10;
end;
